Data-Generating Process

FCMs as a Measure of Stress

FCM Data

Hunting Events

Movement Data

• Contains the location of the 40 collared deer from Feb. 2020 to Feb. 2023.
• Movement is tracked at hourly intervals.

Limited Data, Large Uncertainty

• Hunting events are single points in time and space.
• Deer locations at hourly intervals \(\Rightarrow\) exact distances unknown \(\Rightarrow\) approximate needed, large uncertainty!
• Each deer only encountered few hunting events.

Distance Approximation

Deer location at the time of hunting event is approximated by linear interpolation:

Relevant Hunting Events

The Most Relevant Hunting Event

The Scoring Function

we define the Scoring function as following:

\[ S(d, t) \propto \begin{cases} \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{N}(\mu, \sigma^2) &|t \leq \mu \\ \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{Laplace}(\mu, b) &|t > \mu \end{cases} \] where:

\[ \begin{align*} d & \text{: Distance } \\ t & \text{: Time Difference } \\ \mu & \text{: GRT target = 19 hours } \end{align*} \]

The Scoring Function

The marginal effects of distance and elapsed time since challenge on the score:

The Models

For Modelling, we consider the following covariates, defined for each pair of FCM sample and most relevant hunting event:

The Models

We chose two different approaches to Modelling:

1. Machine Learning: a model, which focuses on prediction, in our case a XGBoost Model
2. Statistical Modelling: a model, which helps to understand the effects of our covariables, here a General Additive Mixed Model

A. XGBoost Setup

1. Split Data: We divide each dataset into training and testing sets. (75% - 25%)
2. Set Hyperparameter Grid: We define a range of values for starting hyperparameters.
3. Optimize Hyperparameters: We then perform a grid search to evaluate different hyperparameter combinations -> Iteratively readjust the grid based on test RMSE until convergence. Uses CV
4. Train Final Model on Full Data: We then use the optimized parameters to train on the entire dataset. We prevent overfitting by optimal n_rounds to keep Test-RMSE low.
5. Aggregate Results: Run the pipeline 40 times with different seeds. Use the average RMSE and average predictions to evaluate the overall performance.

We do this seperately for all 3 datasets (nearest, closest and score).

A. XGBoost Result

B. Generalized Additive Mixed Model

• Family: Gamma

• Log link for interpretability

• Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of faecal samples for each deer

\[ \begin{eqnarray} \textup{FCM}_{ij} &\overset{\mathrm{iid}}{\sim}& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \quad\text{for}\; j = 1,\dots,n_i, \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}), \\ \eta_{ij} &=& \beta_0 + \\ && \beta_1 \cdot \textup{number of other relevant hunting events}_{ij} + \\ && f_1(\textup{time difference}_{ij}) + f_2(\textup{distance}_{ij}) + \\ && f_3(\textup{sample delay}_{ij}) + f_4(\textup{defecation day}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\overset{\mathrm{iid}}{\sim}& \mathcal{N}(0, \sigma_\gamma^2) \quad\text{for} i = 1,\dots,N \end{eqnarray} \]

\(f_1, f_2, f_3, f_4\) are penalised cubic regression splines.

B Generalized Additive Mixed Model

Closest in time

B Generalized Additive Mixed Model

Nearest

B Generalized Additive Mixed Model

Highest score

B Results